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Planes: distances and angles

Problem 9.882 · easy

Find an equation of the plane through \( \displaystyle (0, -3, 2) \) with normal vector \( \displaystyle \left\langle -1, -1, -4 \right\rangle \).

n · (r − P) = 0, i.e. ax + by + cz = n · P.
\[ \left(-4\right) 2 - 1 \cdot 0 - 1 \left(-3\right) = -5 \]
n · P.✓ Proved

Answer \( - x - y - 4 z = -5 \)

Lines: 1 proved, 1 not checked. The answer was also checked a second way, without looking at the solution. The reviewers disagree about how one step is explained; every verdict is in the receipt. How we verify

The full receipt

Line	Status	Checked by	Detail
1	Not checked	—	a sentence; read, not computed
2	✓ Proved	sympy 1.14.0	simplify(lhs - rhs) reduced to 0
answer, a second way	✓ Checked independently	sympy 1.14.0 + mpmath 1.3.0	the plane holds P and its gradient is parallel to n

Reviewers

gpt-oss:20b: pass
qwen3.6:27b-mlx: fail (error) — The solution calculates n · P = -5 but fails to write the final equation of the plane. It stops at the scalar value, omitting the crucial step of equating ax + by + cz to this value.

Every verdict on record (6)

qwen3.6:27b-mlx: fail (error) 2026-11-05 — The solution calculates n · P = -5 but fails to write the final equation of the plane. It stops at the scalar value, omitting the crucial step of equating ax + by + cz to this value.
gpt-oss:20b: pass 2026-11-05
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly identifies the scalar equation of a plane using the normal vector and a point on the plane. The calculation of the constant term is correct.
gpt-oss:20b: pass 2026-11-04
qwen3.6:27b-mlx: pass 2026-11-04 — The solution correctly identifies the scalar equation of a plane using the normal vector and a point on the plane. The calculation of the constant term is correct.
gpt-oss:20b: pass 2026-11-04

Proved: SymPy reduced the difference between the two sides to zero. Checked independently: a separate method, named above, confirmed it. Checked numerically: the two sides agree at every sampled point, which is evidence, not proof. Reviewed: a model or a person read it; that is all a sentence can have. Solution by generator:structured/planes_distance_angle, checked 2026-11-05 with SymPy 1.14.0.